<h2>Problem 167</h2>
<div style="color:#666;font-size:80%;">09 November 2007</div><br />
<div class="problem_content">
<p>For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> = a, U(a,b)<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> = b and for k > 2,
U(a,b)<img src="" style="display:none;" alt="_(" /><sub>k</sub><img src="" style="display:none;" alt=")" /> is the smallest integer greater than U(a,b)<img src="" style="display:none;" alt="_(" /><sub>(k-1)</sub><img src="" style="display:none;" alt=")" /> which can be written in exactly one way as the sum of two distinct previous members of U(a,b).</p>
<p>For example, the sequence U(1,2) begins with<br />
1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, 11 = 3 + 8;<br />
5 does not belong to it because 5 = 1 + 4 = 2 + 3 has two representations as the sum of two previous members, likewise 7 = 1 + 6 = 3 + 4.</p>
<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' />U(2,2<var>n</var>+1)<img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" /> for 2 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' />10, where <var>k</var> = 10<img src="" style="display:none;" alt="^(" /><sup>11</sup><img src="" style="display:none;" alt=")" />.</p>
</div><br />
